3.562 \(\int \sqrt{2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}} \, dx\)

Optimal. Leaf size=233 \[ \frac{4}{17} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{15/2}+\frac{300}{13} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{13/2}-\frac{760}{11} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{11/2}+\frac{304}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{9/2}-\frac{480}{7} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{7/2}+\frac{136}{5} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{5/2}-\frac{16}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \]

[Out]

(-16*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2))/3 + (136*(2 + Sqrt[3 + Sqrt[-1
+ 2*Sqrt[x]]])^(5/2))/5 - (480*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(7/2))/7 + (
304*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(9/2))/3 - (760*(2 + Sqrt[3 + Sqrt[-1 +
2*Sqrt[x]]])^(11/2))/11 + (300*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(13/2))/13
- (56*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(15/2))/15 + (4*(2 + Sqrt[3 + Sqrt[-1
+ 2*Sqrt[x]]])^(17/2))/17

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Rubi [A]  time = 0.604226, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{4}{17} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{15/2}+\frac{300}{13} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{13/2}-\frac{760}{11} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{11/2}+\frac{304}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{9/2}-\frac{480}{7} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{7/2}+\frac{136}{5} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{5/2}-\frac{16}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]

[Out]

(-16*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2))/3 + (136*(2 + Sqrt[3 + Sqrt[-1
+ 2*Sqrt[x]]])^(5/2))/5 - (480*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(7/2))/7 + (
304*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(9/2))/3 - (760*(2 + Sqrt[3 + Sqrt[-1 +
2*Sqrt[x]]])^(11/2))/11 + (300*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(13/2))/13
- (56*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(15/2))/15 + (4*(2 + Sqrt[3 + Sqrt[-1
+ 2*Sqrt[x]]])^(17/2))/17

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Rubi in Sympy [A]  time = 19.3475, size = 202, normalized size = 0.87 \[ \frac{4 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{17}{2}}}{17} - \frac{56 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{15}{2}}}{15} + \frac{300 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{13}{2}}}{13} - \frac{760 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{11}{2}}}{11} + \frac{304 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{9}{2}}}{3} - \frac{480 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{7}{2}}}{7} + \frac{136 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{5}{2}}}{5} - \frac{16 \left (\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+(3+(-1+2*x**(1/2))**(1/2))**(1/2))**(1/2),x)

[Out]

4*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(17/2)/17 - 56*(sqrt(sqrt(2*sqrt(x) - 1)
+ 3) + 2)**(15/2)/15 + 300*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(13/2)/13 - 760*
(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(11/2)/11 + 304*(sqrt(sqrt(2*sqrt(x) - 1) +
3) + 2)**(9/2)/3 - 480*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(7/2)/7 + 136*(sqrt
(sqrt(2*sqrt(x) - 1) + 3) + 2)**(5/2)/5 - 16*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)
**(3/2)/3

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Mathematica [A]  time = 0.163324, size = 183, normalized size = 0.79 \[ \frac{8 \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \left (7 \sqrt{x} \left (2145 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}+1452 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4004 \sqrt{2 \sqrt{x}-1}-3576\right )+4 \left (3843 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}-2535 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4286 \sqrt{2 \sqrt{x}-1}-9786\right )\right )}{255255} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]

[Out]

(8*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2)*(4*(-9786 - 2535*Sqrt[3 + Sqrt[-1
+ 2*Sqrt[x]]] - 4286*Sqrt[-1 + 2*Sqrt[x]] + 3843*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*
Sqrt[-1 + 2*Sqrt[x]]) + 7*(-3576 + 1452*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]] - 4004*Sq
rt[-1 + 2*Sqrt[x]] + 2145*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]])*S
qrt[x]))/255255

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Maple [A]  time = 0.025, size = 154, normalized size = 0.7 \[ -{\frac{16}{3} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{3}{2}}}}+{\frac{136}{5} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{5}{2}}}}-{\frac{480}{7} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{7}{2}}}}+{\frac{304}{3} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{9}{2}}}}-{\frac{760}{11} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{11}{2}}}}+{\frac{300}{13} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{13}{2}}}}-{\frac{56}{15} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{15}{2}}}}+{\frac{4}{17} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x)

[Out]

-16/3*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(3/2)+136/5*(2+(3+(-1+2*x^(1/2))^(1/2))
^(1/2))^(5/2)-480/7*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(7/2)+304/3*(2+(3+(-1+2*x
^(1/2))^(1/2))^(1/2))^(9/2)-760/11*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(11/2)+300
/13*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(13/2)-56/15*(2+(3+(-1+2*x^(1/2))^(1/2))^
(1/2))^(15/2)+4/17*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(17/2)

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Maxima [A]  time = 0.727288, size = 207, normalized size = 0.89 \[ \frac{4}{17} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{17}{2}} - \frac{56}{15} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{15}{2}} + \frac{300}{13} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{13}{2}} - \frac{760}{11} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{11}{2}} + \frac{304}{3} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{9}{2}} - \frac{480}{7} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{7}{2}} + \frac{136}{5} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{5}{2}} - \frac{16}{3} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2),x, algorithm="maxima")

[Out]

4/17*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(17/2) - 56/15*(sqrt(sqrt(2*sqrt(x) - 1
) + 3) + 2)^(15/2) + 300/13*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(13/2) - 760/11*
(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(11/2) + 304/3*(sqrt(sqrt(2*sqrt(x) - 1) + 3
) + 2)^(9/2) - 480/7*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(7/2) + 136/5*(sqrt(sqr
t(2*sqrt(x) - 1) + 3) + 2)^(5/2) - 16/3*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(3/2
)

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Fricas [A]  time = 0.272282, size = 115, normalized size = 0.49 \[ -\frac{8}{255255} \,{\left ({\left (847 \, \sqrt{x} - 1688\right )} \sqrt{2 \, \sqrt{x} - 1} - 2 \,{\left ({\left (1001 \, \sqrt{x} + 6800\right )} \sqrt{2 \, \sqrt{x} - 1} - 2352 \, \sqrt{x} - 29712\right )} \sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} - 30030 \, x + 3843 \, \sqrt{x} + 124080\right )} \sqrt{\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2),x, algorithm="fricas")

[Out]

-8/255255*((847*sqrt(x) - 1688)*sqrt(2*sqrt(x) - 1) - 2*((1001*sqrt(x) + 6800)*s
qrt(2*sqrt(x) - 1) - 2352*sqrt(x) - 29712)*sqrt(sqrt(2*sqrt(x) - 1) + 3) - 30030
*x + 3843*sqrt(x) + 124080)*sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+(3+(-1+2*x**(1/2))**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2),x, algorithm="giac")

[Out]

Timed out